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<title>FFTW 3.3.8: The Discrete Hartley Transform</title>

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<a name="The-Discrete-Hartley-Transform"></a>
<div class="header">
<p>
Previous: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="p" rel="prev">Real even/odd DFTs (cosine/sine transforms)</a>, Up: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="u" rel="up">More DFTs of Real Data</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="The-Discrete-Hartley-Transform-1"></a>
<h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>

<p>If you are planning to use the DHT because you&rsquo;ve heard that it is
&ldquo;faster&rdquo; than the DFT (FFT), <strong>stop here</strong>.  The DHT is not
faster than the DFT.  That story is an old but enduring misconception
that was debunked in 1987.
</p>
<p>The discrete Hartley transform (DHT) is an invertible linear transform
closely related to the DFT.  In the DFT, one multiplies each input by
<em>cos - i * sin</em> (a complex exponential), whereas in the DHT each
input is multiplied by simply <em>cos + sin</em>.  Thus, the DHT
transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
convenient property of being its own inverse.  In FFTW, a DHT (of any
positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
<a name="index-FFTW_005fDHT"></a>
<a name="index-discrete-Hartley-transform"></a>
<a name="index-DHT"></a>
</p>
<p>Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
size <code>n</code> followed by another DHT of the same size will result in
the original array multiplied by <code>n</code>.
<a name="index-normalization-4"></a>
</p>
<p>The DHT was originally proposed as a more efficient alternative to the
DFT for real data, but it was subsequently shown that a specialized DFT
(such as FFTW&rsquo;s r2hc or r2c transforms) could be just as fast.  In FFTW,
the DHT is actually computed by post-processing an r2hc transform, so
there is ordinarily no reason to prefer it from a performance
perspective.<a name="DOCF5" href="#FOOT5"><sup>5</sup></a>
However, we have heard rumors that the DHT might be the most appropriate
transform in its own right for certain applications, and we would be
very interested to hear from anyone who finds it useful.
</p>
<p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
multi-dimensional transform, FFTW computes the separable product of 1d
DHTs along each dimension.  Unfortunately, this is not quite the same
thing as a true multi-dimensional DHT; you can compute the latter, if
necessary, with at most <code>rank-1</code> post-processing passes
[see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264&ndash;266 (1987)].
</p>
<p>For the precise mathematical definition of the DHT as used by FFTW, see
<a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>.
</p>
<div class="footnote">
<hr>
<h4 class="footnotes-heading">Footnotes</h4>

<h3><a name="FOOT5" href="#DOCF5">(5)</a></h3>
<p>We provide the DHT mainly as a byproduct of some
internal algorithms. FFTW computes a real input/output DFT of
<em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
and then using Rader&rsquo;s prime-DFT algorithm adapted to the DHT.</p>
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Previous: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="p" rel="prev">Real even/odd DFTs (cosine/sine transforms)</a>, Up: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="u" rel="up">More DFTs of Real Data</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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